90 IV ADDITIONAL PROPERTIES OF THE REAL LINE

be such that for y e A n E(x; r\ \f(y) -/(x)| < e. Let C = A n B(x; r),
D = A — C; if x' e E - A and d(x, x') ^ r/4, we have, for each y e D,
d(x'9 y) ^ d(x, y) - d(x, x') > 3r/4, hence

yeD

on the other hand, f(x) d(x'9 x) < 2d(;c', x) ^ r/2, and therefore
inf (/GO 4*', j;))

yeA yeC

But, as /(x) - £ </GO </(*) + e for j> e C, and inf d(jc', y) = rf(x', A),
we have ye€

(f(x) - e) dfjc', A) < inf (f(y) d(x', y)) ^ (/(x) + e) d(x'9 A)

which proves that \g(xr) —f(x)\ ^ fi for x' e E — A and d(x9 x') < r/4; on
the other hand, if x' e A and d(x, *') < r/4, |^(xr) -/(x)| = \f(x') -f(x)\ ^ e,
and this ends the proof.

(4.5.2) Let A, B be two nonempty closed sets in a metric space E, such that
A n B = 0. Then there is a continuous function f defined in E, with values
in [0, 1], such that f(x) = I in A andf(x) = 0 in B.

Apply (4.5.1) to the mapping of A u B in R, equal to 0 in B and to 1
in A, which is continuous in A u B.

PROBLEMS

1. In a metric space E, let (Fn) be a sequence of closed sets, A the union of the Fn; if
x $ A, show that there exists a bounded continuous function /> 0 defined in E, such
that /(jc) = 0 and f(y) > 0 for each y e A (use (4.5.2) and (7.2.1)).

2. (a) Let E be a metric space such that every bounded set in E is relatively compact;
show that E is locally compact and separable (use (3.16.2)).

(b) Conversely, let E be a locally compact, noncompact separable metric space, d the
distance on E; let (Un) be a sequence of relatively compact open subsets of E such that
On ^ Un+l and E is the union of the sequence (Un) (3.18.3). Show that there exists a
continuous real-valued function / in E such that f(x) ^ n for x e Ort and f(x) ^ n for
x e E - U« (use (4.5.2)); the distance d'(x, y) = d(x, y) + \f(x) -f(y)\ is then topologi-
cally equivalent to d, and for d', any bounded set is relatively compact.Let r = d(x, A); for